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Actually, the more likely wire that's failing the Green/Black wire that carries the signal from the sender in the engine sump to the gauge.

If I were testing this issue, I would make a long jumper wire and simply disconnect the sender, and install the jumper wire, then run it out from under the car and in the door and over to the gauge and connect it there (after removing the Grn/Blk). See if that solves the problem, then case the Green/Black wire to find the failure. It's probably rubbed through the insulation somewhere near the motor. There is a connector under the battery (usually in a plastic housing) and I'd inspect the length of wire from that connector to the sump sender first.